Crystal families and systems in higher dimensions, and geometrical symbols of their point groups. II. Cubic families in five- and n-dimensional spaces.
This paper is devoted to the study of the crystal families with cubic symmetries and to the mathematical construction of all their point-symmetry groups. The mono cubic crystal families of n-dimensional space (E(n)) are defined and a list of these families is given for spaces E4, E5, E6 and E7 with the Weigel-Phan-Veysseyre (WPV) symbols of their holohedries. The cubic and iso cubic crystal point-symmetry groups of space E(n) are also defined together with their properties and their WPV symbols. Some examples of these point groups are given. The 16 point groups of the three isomorphic mono cubic crystal families, the cubic-al family of space E4 (No. 19), the cube oblique (or cube parallelogram) family of space E5 (No. XVIII) and the triclinic cubic family of space E6 (No. 21) are listed. All the WPV symbols of the point-symmetry groups of all the mono cubic crystal families of space E5, i.e. the cube rectangle family (No. XXII), the cube square family (No. XXVI) and the cube hexagon family (No. XXVII), are given together with an explanation of the mathematical construction of these point-symmetry groups. All the di cubic crystal families of spaces E6, E7 and E8 are predicted and the symbols of their holohedries are given. Finally, some tri cubic crystal families of spaces E9, E10 and E11 are listed.
Weigel
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Laboratoire Mathématiques Appliquées aux Systèmes, Ecole Centrale Paris, Grande Voie des Vignes, 92295 Châtenay-Malabry Cedex, France.
Phan
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Veysseyre
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